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Lipschitz Estimates in Almost‐Periodic Homogenization
Author(s) -
Armstrong Scott N.,
Shen Zhongwei
Publication year - 2016
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21616
Subject(s) - mathematics , lipschitz continuity , neumann boundary condition , bounded function , homogenization (climate) , constructive , dirichlet distribution , mathematical analysis , boundary (topology) , divergence (linguistics) , lipschitz domain , boundary value problem , computer science , biodiversity , ecology , linguistics , philosophy , process (computing) , biology , operating system
We establish uniform Lipschitz estimates for second‐order elliptic systems in divergence form with rapidly oscillating, almost‐periodic coefficients. We give interior estimates as well as estimates up to the boundary in bounded C 1, α domains with either Dirichlet or Neumann data. The main results extend those in the periodic setting due to Avellaneda and Lin for interior and Dirichlet boundary estimates and later Kenig, Lin, and Shen for the Neumann boundary conditions. In contrast to these papers, our arguments are constructive (and thus the constants are in principle computable) and the results for the Neumann conditions are new even in the periodic setting, since we can treat nonsymmetric coefficients. We also obtain uniform W 1,p estimates.© 2016 Wiley Periodicals, Inc.